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Sigma-additive
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ba Space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma. The norm is defined as the variation, that is \, \nu\, =, \nu, (X). If Σ is a sigma-algebra, then the space ca(\Sigma) is defined as the subset of ba(\Sigma) consisting of countably additive measures. The notation ''ba'' is a mnemonic for ''bounded additive'' and ''ca'' is short for ''countably additive''. If ''X'' is a topological space, and Σ is the sigma-algebra of Borel sets in ''X'', then rca(X) is the subspace of ca(\Sigma) consisting of all regular Borel measures on ''X''. Properties All three spaces are complete (they are Banach spaces) with respect to the same norm defined by the total variation, and thus ca(\Sigma) is a closed subset of ba(\Sigma), and rca(X) is a closed set of ca(\Sigma) for Σ the algebra of Borel sets on ''X''. The space of simple functions on \Sigma is dense in ba(\Sigma). T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma-additive Set Function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R and \pm \infty. A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. Definitions If \mathcal is a family of sets over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal and codomain \infty, \infty/math> or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The domain is a set function may have any number properties; the commonly encountered properties and categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Set Function
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely-additive set function (the terms are equivalent). However, a finitely-additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are abstrac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . The notation , meaning the set of all functions from S to a given set of two elements (e.g., ), is used because the powerset of can be identified with, equivalent to, or bijective to the set of all the functions from to the given two elements set. Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (geometry)
In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set X with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space \R^n. Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra. Definition Let X be a set and \mathcal S be a collection of admissible subsets of X. A function \phi on \mathcal S with values in an abelian semigroup R is called a valuation if it satisfie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subadditive Set Function
In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions. Definition Let \Omega be a set and f \colon 2^ \rightarrow \mathbb be a set function, where 2^\Omega denotes the power set of \Omega. The function ''f'' is ''subadditive'' if for each subset S and T of \Omega, we have f(S) + f(T) \geq f(S \cup T). Examples of subadditive functions Every non-negative submodular set function is subadditive (the family of non-negative submodular functions is strictly contained in the family of subadditive functions). The function that counts the number of sets required to cover a given set is subadditive. Let T_1, \dotsc, T_m \subseteq \Omega such that \cup_^m T_i=\Omega. Define f as the minimum number of subsets required to cover a given set. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be Holomorphic function, holomorphic in the upper half-plane (among other requirements). Instead, modular functions are Meromorphic function, meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Measure
In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures". Given a measurable space (X, \Sigma) (that is, a set X with a σ-algebra \Sigma on it), an extended signed measure is a set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ... \mu : \Sigma \to \R \cup \ such that \mu(\varnothing) = 0 and \mu is sigma additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
